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### Probability of a Straight Flush

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Date: 5/15/96 at 14:44:22
From: Art Mabbott
Subject: Probability Problem

My high school math topics class is trying to work through a
probability problem involving counting and trees.  The question
involves finding the probability of drawing 5 spades, that is
P(a flush) = (13 12 11 10 9)/(52 51 50 49 48) P(flush of any suit)
= 4 P(a flush)

The question is, what is the P(Straight flush) = ?  (knowing that
there are only 10 ways in each suit to draw a straight flush).

Art Mabbott
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Date: 5/17/96 at 19:0:51
From: Doctor Ken
Subject: Re: Probability Problem

Hello!

Something that will help in these problems is the "choose" formula.
If you want to know the number of ways you can choose 5 cards from a
deck of 52 cards, when the order of the cards doesn't matter, then the
answer is 52 choose 5, which is

52!/(5!(52-5)!) = (52 51 50 49 48)/(5 4 3 2 1).

So that's the total number of different hands you could be dealt.

Since you've already figured out how many different ways you can get a
straight flush (actually, are you sure it's 10 in each suit?  I get 9,
because A2345 doesn't count, right?), you can just divide the number
of straight flushes by the number of possible hands to get the
probability you're dealt a straight flush.

-Doctor Ken,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
High School Probability

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